Proof : That index(I) = m is computed by R. T. Smith [SRT]. ..... the smoothness of the transition maps, let (W1, Î1), (W2, Î2) be two charts corresponding to F1 ...
Harmonic Maps with Fixed Singular Sets
ROBERT HARDT* and LIBIN MOU
§ 0. Introduction Here, for a smooth domain Ω in R and a compact smooth Riemannian manifold N we study a space H consisting of all harmonic maps u : Ω → N that have a singular set being a fixed compact subset Z of Ω having finite m − 3 dimensional Minkowski content. This holds if, for example, Z is m − 3 rectifiable [F, 3.2.14]. We define a suitable topology on H using H¨older norms on derivatives weighted by powers of the distance to Z. We study the structure of H and other properties, such as stability and minimality, under perturbations. For example, near a homogeneous harmonic map that is singular only at the origin, H is, for suitable powers of the weights, a smooth manifold [Theorem 4.7]. An application [Theorem 5.6] of the result to some well-known harmonic maps, such as the homogeneous extension of the harmonic maps from S2 to S2 , the identity from Sm−1 to itself and the equator maps from Sm−1 to Sm , gives that their boundary data can be perturbed in any directions of the eigenvectors corresponding to the positive eigenvalues (of the boundary data) to obtain new harmonic maps. These new harmonic maps are also stable (energy minimizing, unique) if the perturbed maps are strictly stable (strictly minimizing), which is implied by Theorems 3.5, 3.8 saying that stability and minimality are preserved under small perturbations. For smooth harmonic maps, we prove that H is a Banach manifold modelled on the space of boundary data, and the projection map that sends u to u|∂Ω is Fredholm of index 0 [Theorem 6.4]. Locally, near u ∈ H with K0 being the space of smooth Jacobi fields along u that vanish on ∂Ω, some neighborhood of u in H is diffeomorphic to a submanifold (of codimension dim (K0 )) of the product of K0 with the space of the boundary data [Theorem 6.2]. Using the global structure theorem, we prove a generic uniqueness property of smooth harmonic maps [Theorem 6.8], which establishes the first category nature of the set of all boundary maps that occur as trace of two distinct harmonic maps having same energy. We prove a Schauder-type estimate and a pointwise estimate in Section 2 [Theorems 2.1, 2.3]. This is relevant for a study of the Fredholm property of the Jacobi operator, which is essential to apply the implicit function theorem. For isolated singularities, a very detailed analysis of this property with perturbation applications has recently been given by N. Smale [SN3]. If one allows the singularity to vary, there is the example of [HKL] where the boundary data corresponds under the sterographic projection to the conformal map z 2 . Here there is a one parameter family of distinct harmonic maps having this Dirichlet boundary data and the same energy (by energy minimality). The singularity varies. This phenomenon holds for any n-axially symmetric boundary data of non-zero degree when n ≥ 2. (See [HKL].) For smooth case, we follow with some modifications some steps in Brian White’s study [WB] of the corresponding theory for smooth immersed minimal submanifolds. The presence of singularities demands various new estimates and leads to several interesting examples of singular Jacobi fields and families of harmonic maps. The authors would like to thank Frank Pacard and Nat Smale for their interest and for pointing out an error in an earlier version of this paper. m
§ 1. Preliminaries Appeared in Journal of Geometric Analysis. 1
Vol.
2, 5 (1992) 445-488.
1.1. Definitions. Here we assume that Ω ⊆ Rm (m ≥ 3) is a smooth bounded domain with standard metric, and N ⊆ Rp (p ≥ 2) is a smooth compact submanifold (without boundary). All our discussions can be modified to hold with Ω replaced by a compact Riemannian manifold with boundary. Let H 1 (Ω, N ) = {u ∈ H 1 (Ω, Rp ) : u(x) ∈ N for a.e. x ∈ Ω} where H 1 (Ω, Rp ) is the set of u ∈ L2 (Ω, Rp ) whose distributional gradient is square integrable. The energy of u ∈ H 1 (Ω, Rp ) is denoted Z E(u) = |∇u|2 . Ω
Now we introduce the spaces for maps that are possibly singular (i.e., discontinuous) on a fixed subset of Ω. Let Z be a compact subset of Ω having finite m − q (q ≥ 0) dimensional Minkowski content Mm−q (Z) [F, 3.2.37]. For x ∈ Ω, denote by ρ(x) = dist(x, Z), and for r > 0, denote Zr = {z ∈ Rm : ρ(z) < r};
Zr,2r = Z2r \ Z¯r ;
Ω0 = Ω \ Z;
Ωr = Ω \ Z¯r .
Take a number r0 > 0 so that Z3r0 ⊆ Ω. For an integer k ≥ 0, 0 ≤ α < 1 and ν ∈ R, we define a norm k,α kukk,α;ν for u ∈ Cloc (Ω0 , Rp ): kukk,α;ν = sup
0 2 and ν ≥ 0, then C k,α;ν (Ω0 , Rp ) ⊂ H 1;0 (Ω, Rp ) ⊂ H 1 (Ω, Rp ). Proof : We claim that if ρ(x) = dist(x, Z) and ν + q > 0, then ρν ∈ L1 (Ω).
(1.3.1)
To prove (1.3.1), it suffices to show that ρν ∈ L1 (Zr0 ) for small r0 . Indeed, that Mm−q (Z) is finite implies that for some constant C1 and small r0 , Lm (Zr ) ≤ C1 rq when 0 < r < r0 . Thus we have Z ∞ Z X ρν dx = ρν dx Zr0
i=0
≤
∞ X
Z2−i−1 r
0 ,2
−i r 0
((r0 2−i−1 )ν + (r0 2−i )ν )(r0 2−i )q
i=0
= (1 + 2−ν )r0ν+q
∞ X
(2ν+q )−i
i=0
(1 + 2 )r0ν+q 1 − 2−ν−q −ν
=
< ∞.
Now that δ < ν + q−2 implies that if u ∈ C k,α;ν (Ω0 , Tu N ), then for j = 0, 1, ..., k, |ρj−δ−1 ∇j u|2 ≤ 2 1 Cρ ∈ L . By definition, then u ∈ H k;δ (Ω, Rp ). 2(ν−δ−1)
3
In the first and second variational formulae that follow, we assume that k ≥ 2, 0 ≤ α < 1 and ν ≥ 0. 1.4. First variation formula of energy. Since N is smooth, we may take a δ = δ(N ) > 0 such that for any x ∈ Nδ ≡ {x ∈ Rp : dist(x, N ) ≤ δ}, there exists a unique point π(x) ∈ N such that dist(x, N ) = |x − π(x)|. Furthermore, for y ∈ N , (1.4.1)
Dπ(y) = Py ,
the projection map from Rp to Ty N [HL1]. Suppose u ∈ C k,α;0 (Ω0 , N ), κ ∈ H 1 (Ω, Tu N ) ∩ L∞ and ut = π(u + tκ) : (−ε, ε) → H 1 (Ω, N ) is a d u = Dπ(u)κ = Pu κ = κ ∈ Tu N , then the first variation differentiable curve in H 1 (Ω, N ) with u = u0 , dt t=0 t formula is Z Z d ∂u −H(u) · κ + · κ, E(ut ) = 2 dt t=0 ∂n Ω ∂Ω where H(u) = Pu ∆u = ∆u − A(u) (∇u, ∇u), n is the outward normal unit direction of ∂Ω, A(u) is the second fundamental form of N , evaluated at u, ∂u ∂u ∂u and ∇u = ( ∂x , ..., ∂x ). Note that ∂x ∈ Tu N for each i. Also H(u) ∈ C k−2,α;−2 (Ω0 , Tu N ) and 1 m i H : C k,α;0 (Ω0 , N ) → C k−2,α;−2 (Ω0 , Rp ) is a smooth map (between the two Banach spaces). A map u ∈ H 1 (Ω, N ) is a harmonic map if it is a solution of Euler-Lagrange equation (1.4.2)
H(u) = 0.
In this paper we are interested in set of harmonic maps in C k,α;0 (Ω0 , N ), which we denote by H = {u ∈ C k,α;0 (Ω0 , N ) : u is harmonic}. 1.5. Remarks . (a). H(u) can be viewed as the derivative of E(u): For κ ∈ H 1 (Ω, Tu N ) ∩ L∞ , Z hκ, DE(u)i = Ω
−H(u) · κ = h−H(u), κiL2 .
(b). Also H(u)=Trace (∇du), where ∇du is the bilinear form defined by ∇du(κ, η) = (∇κ du)(η). See [EL][J]. (c). H(u) can also be described in terms of local coordinates of N . See [J], for example. 1.6. Second variational formula of energy. Suppose that u ∈ C k,α;0 (Ω0 , N ), κ, η ∈ H 1 (Ω, Tu N ) ∩ L∞ and ust : (−ε, ε) × (−ε, ε) → H 1 (Ω, N ) (e.g., ust = π(u + tκ + sη)) is a differentiable 2-parameter variation with (κ, η) = (
∂ ∂ ust , ust ) . ∂t ∂s s=t=0 4
Then the second variation is Z Z ∂ 2 ∂κ E(ust ) = 2 −Ju κ · η + 2 · η, ∂t∂s s=t=0 Ω ∂Ω ∂n
(1.6.1)
where Ju = Pu ◦ DH(u) is given by (1.6.2)
Ju κ = ∆κ − DA(u)(κ, ∇u, ∇u) − 2A(u)(∇κ, ∇u) + A(u)(H(u), κ).
1.7 . Ju is called the Jacobi operator with respect to u. Note that Ju maps C k,α;ν (Ω0 , Tu N ) to C k−2,α;ν−2 (Ω0 , Tu N ), and it is linear and uniformly elliptic. A solution κ ∈ C k,α;ν (Ω0 , Tu N ) of Ju κ = 0 is called a Jacobi field in C k,α;ν (Ω0 , Tu N ) with respect to u. Furthermore, the symmetry of κ and η in (1.6.1) implies that Ju is self-adjoint; that is, for all κ, η ∈ H01 (Ω, Tu N ) ∩ L∞ , Z 2 2 (1.7.1) hJu κ, ηiL = hκ, Ju ηiL = − ∇κ · ∇η + DA(κ, ∇u, ∇u) · η. Ω
If u is harmonic, then (1.7.2)
Ju κ = hκ, DH(u)i = ∆κ − DA(u)(κ, du, du) − 2A(u)(dκ, du).
If u is harmonic, then for κ ∈ H 1 (Ω, Tu N ) ∩ L∞ , 1 Ju κ = hκ, DH(u)i = lim [H(π(u + tκ)) − H(u)]. t→0 t This extends the definition of Ju to H 1 (Ω, Rp ) so that Ju κ = 0 for κ⊥ Tu N . Ju may have different forms. See [J], for example. § 2. Some Estimates for Jacobi Fields To use implicit function theorem to study the solutions of H(u) = 0, a key step is to show the Fredholm property of the linearization of H, (i.e., the Jacobi operator Ju ). For this purpose and other possible applications, we prove an estimate of Schauder’s type and a pointwise bound estimate for solutions in C k,α;ν (Ω0 , Rp ) of elliptic systems whose coefficients may have singularities. We will use the notations introduced in Section 1, with k ≥ 2, 0 < α < 1 and ν ∈ R. k,α 2.1. Theorem. Suppose κ ∈ Cloc (Ω0 , Rp ) and J is a linear elliptic operator in the following form:
Ju =
X p
Jij (·, D)uj
j=1
Jij (x, D) =
X
, 1≤i≤p
ast ij (x)Ds Dt +
X
bsij (x)Ds + cij (x),
s
st
where the coefficients satisfy the following: 2
k−2,α;0 a = (ast (Ω0 , Rp ij ) ∈ C
×m2
), b = (bsij ) ∈ C k−2,α;−1 (Ω0 , Rp
2
c = (cij ) ∈ C k−2,α;−2 (Ω0 , Rp ), Jκ ∈ C k−2,α;ν−2 (Ω0 , Rp ), X J(x, ξ) = det ast ≥ λ |ξ|2p , ij ξs ξt st
5
p×p
2
×m
),
for all ξ ∈ Rm and some constant λ > 0. Then there exists a constant C depending only kakk−2,α;0 , kbkk−2,α;−1 , kckk−2,α;−2 , ν, m, p, λ and Ω such that h i kκkk,α;ν ≤ C kJκkk−2,α;ν−2 + kκkk−2,0;ν + kκkk,α,∂Ω .
(2.1.1)
Proof : For 0 < r < r0 , and x0 ∈ Zr,2r , we apply the Schauder’s interior estimate on the ball BR = BR (x0 ) with R = 2r/3. To be specific, let σ ∈ R and consider the following norms: (2.1.2)
(σ)
kκkk,α;BR =
k X
j=0 x∈BR
∗
(2.1.3)
|∇j κ(x)| + sup dj+σ x
k X
j=0 x,y∈BR
k X X
(σ)
kκkk,α;BR =
sup dj+α+σ xy
|∇j κ(x) − ∇j κ(y)| , |x − y|α
Rj+β+σ k∇j κk(β),BR ,
β∈{0,α} j=0 (σ)
(σ)
where dx = dist(x, ∂BR ) and dxy = min{dx , dy }. Here k · kk,α;BR is equivalent to the norm | · |k,α;BR defined in [GT, § 6.1]. When σ ≥ 0, the following simple relation holds ∗
(2.1.4)
(σ)
(σ)
(σ)
kκkk,α;BR/2 ≤ kκkk,α;BR ≤∗ kκkk,α;BR ≤ kκkk,α;−σ .
(where kκkk,α;−σ is the norm defined in 1.1) In particular, we have (0)
kakk−2,α;BR ≤ kakk−2,α;0 , (1)
kbkk−2,α;BR ≤ kbkk−2,α;−1 , (2)
kckk−2,α;BR ≤ kckk−2,α;−2 . Thus the Schauder’s interior estimates [GT , Theorem 6.2] ([DN, Theorem 1] for systems) imply h i (0) (2) kκkk,α;BR ≤ C1 kJκkk−2,α;BR + kκk0,BR ,
(2.1.5)
where C1 , and C2 , C3 ... in the sequel, are constants as stated in the theorem. Combined with (2.1.4), (2.1.5) gives (2.1.6)
k X X
rj+β k∇j κk(β),Br/3 ≤ C2
β∈{0,α} j=0
h X k−2 X
i rj+β+2 k∇j Jκk(β),B2r/3 + kκk0,B2r/3 .
β∈{0,α} j=0
Multiplying (2.1.6) by r−ν , we get (2.1.7)
k X X
rj+β−ν k∇j κk(β),Br/3
β∈{0,α} j=0
≤ C2
h X k−2 X h
i rj+β−(ν−2) k∇j Ju κk(β),B2r/3 + r−ν kκk0,B2r/3 ,
β∈{0,α} j=0
i ≤ C3 kJκkk−2,α;ν−2 + kκkk−2,0;ν . 6
Now we apply the fact that any function η ∈ C α (Ω0 ) satisfies i h kηkα,Zr,2r ≤ 2 · 3α sup r−α kηk0,Br/3 (x) + kηk(α),Br/3 (x) x∈Zr,2r
to ∇j κ (j = 0, · · · , k), and use (2.1.7), to get (2.1.8)
k X X
rj+β−ν k∇j κk(β),Zr,2r
β∈{0,α} j=0
≤ C4 sup
x∈Zr,2r
h
k X X
rj+β−ν k∇j κk(β),Br/3 (x)
β∈{0,α} j=0
i ≤ C5 kJκkk−2,α;ν−2 + kκkk−2,0;ν . To estimate kκkk,α,Ωr0 , we apply the Schauder estimates [ADN, Theorem 9.3], to Ωr0 ⊂ ⊂ Ωr0 /2 (with D = Ωr0 /2 , A = Ωr0 and Γ = ∂Ω) to get h i (2.1.9) kκkk,α,Ωr0 ≤ C6 kJκkk−2,α,Ωr0 /2 + kκkk,α,∂Ω + kκk0,Ωr0 /2 h i ≤ C7 kJκkk−2,α;ν−2 + kκkk,α,∂Ω + kκkk−2,0;ν . Summing up (2.1.8) and (2.1.9), we get the desired estimate. Later we will use the following corollary to Theorem 2.1 with k = 2. 2,α 2.2. Corollary. If κ ∈ Cloc (Ω0 , Tu N ) ∩ C 0,0;ν (Ω0 , Tu N ) is solution of Ju κ = f with κ|∂Ω ∈ C 2,α (∂Ω, N ) and f ∈ C 0,α,ν−2 (Ω0 , Tu N ), then κ is a Jacobi field in C 2,α;ν (Ω0 , Tu N ).
Proof : This is implied by the estimates (2.1.1) in Theorem 2.1: h i kκk2,α;ν ≤ C kκk0,0;ν + kκk2,α,∂Ω + kf k0,α,ν−2 .
In order to apply Corollary 2.2, we need that κ ∈ C 0,0;ν (Ω0 , Tu N ), or, |κ(x)| ≤ Cρν (x) for some constant 2,α C. This may not true for a solution κ ∈ Cloc (Ω0 , Tu N ) of Ju κ = f . On the other hand we have the following estimate. 2,α 2.3. Theorem. Let µ ∈ R, 0 < α < 1 and f ∈ C 0,α,µ (Ω0 , Tu N ). If κ ∈ Cloc (Ω0 , Tu N ) ∩ Lp (Ω, Tu N ), p > 1 and Ju κ = f , then there is a constant C such that for x ∈ Ω0 , h i (2.3.1) |κ(x)| ≤ C ρ(x)−m/p kκkLp (Bρ(x)/2 (x)) + ρ(x)µ+2 kf k0,α,µ ,
where ρ(x) = dist(x, Z). Proof : For x ∈ Ω0 , let R = ρ(x)/4 and v ∈ C01 (BR (x)), v ≥ 0. Then as shown in [ML1, Lemma], we have, Z Z ∆κ · κ (2.3.2) ∇|κ| · ∇v ≤ v. |κ| BR (x) BR (x) Now since Ju κ = 0, by (1.6.2) and the fact that A(u)⊥Tu N , we have (2.3.3)
|∆κ · κ| ≤ |DA(u)(κ, ∇u, ∇u) · κ| + |f ||κ| ≤ C1 |∇u|2 |κ|2 + |f ||κ| 7
for some constant C depending only on N . Now (2.3.2) and (2.3.3) imply that |κ| is a subsolution of ∆w + C1 |∇u|2 w| + C1 |f | = 0,
(2.3.4) in the sense that Z
∇|κ|∇v − C1 |∇u|2 |κ|v − C1 |f |v ≤ 0
(2.3.5) BR (x)
for v ∈ C01 (BR (x)), v ≥ 0. By Theorem 8.17 in [GT], for p > 1 and q > m, there exists a constant C3 depending only on m, p and q such that h i (2.3.6) sup |κ| ≤ C R−m/p kκkLp (B2R (x)) + R2(1−m/q) kf kLq/2 (B2R (x)) . BR (x)
Note that (2.3.7)
kf kLq/2 (B2R (x)) ≤ ρ(x)µ+2m/q kf k0,α,µ .
So (2.3.7) and (2.3.6) imply (2.3.1).
§ 3. Strict Minimality and Stability Under Perturbations The main results of this section are that stability and minimality of harmonic maps are preserved under small perturbations. Here the singular sets can be as general as in Section 1. For the stability of perturbed harmonic maps with isolated singularities, a similar result was proved in [ML1]. For related results on minimal surfaces, see [HS] and [SN1]. As in section 1, let Ω ⊂ Rm be bounded smooth (say C 3 ) domain and N ⊂ Rp is a smooth compact submanifold (say C 4 ), m ≥ 3, p ≥ 3. Let Z be a fixed compact subset of Ω having finite m − 3 dimensional Minkowski content Mm−3 (Z) < ∞ [F, 3.2.37] and Ω0 = Ω\Z. We consider the maps in the space C 1;0 (Ω0 , N ) that is defined by 1 C 1;0 (Ω0 , N ) = {u ∈ Cloc (Ω0 , N ) : kuk1;0 < ∞}, where kuk1;0 = supx∈Ω0 [|u(x)| + ρ(x)|∇u(x)|]. Note that C 1;0 (Ω0 , N ) = C 1,0;0 (Ω0 , N ), defined as in Section 1 with k = 1, α = 0 and ν = 0. Lemma 1.3 implies that C 1;0 (Ω0 , N ) ⊆ H 1;0 (Ω, N ) ⊆ H 1 (Ω, N ). Thus every map u ∈ C 1;0 (Ω0 , N ) has finite energy E(u) =
R Ω
|∇u|2 .
3.1. Definitions. For u ∈ H 1 (Ω, N ), let Hu1 (Ω, N ) = {v ∈ H 1 (Ω, N ) : v = u on ∂Ω}, H01 (Ω, Tu N ) = {ξ ∈ H01 (Ω, Rp ) : ξ ∈ Tu N }. We say that u ∈ C 1;0 (Ω0 , N ) is strictly minimizing (with rate µ) if Z (3.1.2)
E(v) − E(u) ≥ µ Ω
|v − u|2 , ρ2
ρ(x) = dist(x, Z),
for and some µ > 0 and all v ∈ Hu1 (Ω, N ). If µ = 0, then u is minimizing. 8
We say that u is strictly stable (with rate µ) if Z d2 |ξ|2 (3.1.3) E(u ) ≥ 2µ t 2 2 t=0 dt Ω ρ for some µ > 0 and all ξ ∈ H01 (Ω, Tu N ) ∩ L∞ , where ut : (−ε, ε) → H 1 (Ω, N ) is a curve satisfying u0 = u d and dt u = ξ. We say u is stable if µ = 0. t=0 t Using Jacobi operator Ju (define in 1.9), we can rewrite (3.1.3) as Z Z |ξ|2 (3.1.4) − Ju ξ · ξ ≥ 2µ . 2 Ω Ω ρ If (3.1.2) holds for v ∈ Hu1 (Ω, N ) satisfying sup |v − u| ≤ δ for some δ > 0, we say that u is strictly nearby minimizing, or strictly δ-nearby minimizing with rate µ. We now give a formula being used to compare the energies of two maps. It was first shown by Macintosh and Simon [MS] for the case Ω0 = Bm \ {0}. 3.2. Theorem. Suppose that u ∈ C 1;0 (Ω0 , N ) is a harmonic map and v ∈ Hu1 (Ω, N ). Let ξ = v − u. Then Z (3.2.1) E(v) − E(u) = |∇ξ|2 − Su (x, ξ), Ω
where (3.2.2)
Su (x, q) = Qu (x, q, q) + Ru (x, q),
and Qu : Ω0 × Rp × Rp → R is a bilinear form for each x ∈ Ω0 and Ru : Ω0 × Rp → R is a map that satisfies |q|3 (3.2.3) |Ru (x, q)| + ρ(x)|Dx Ru (x, q)| + |q||Dq Ru (x, q)| ≤ C0 2 ρ p for some constant C0 (N, kuk1,0 , n, p), all q ∈ R and x ∈ Ω0 with u(x) + q ∈ N . Proof : For v ∈ Hu1 (Ω, N ), let ξ = v − u. We have, by integration by parts and (1.4.2) Z Z (3.2.4) |∇v|2 − |∇u|2 = |∇ξ|2 + 2∇ξ · ∇u Ω Ω Z Z 2 = |∇ξ| − 2ξ · ∆u = |∇ξ|2 − 2ξ · A(u)(∇u, ∇u). Ω p
Ω
ξy> and p
For y ∈ N and ξ ∈ R , let respectively. Define B : N × Rp → R by
ξy⊥
be the orthogonal projections of ξ into Ty N and (Ty N )⊥ ,
1 B(y, ξ) = ξy⊥ − A(y)(ξy> , ξy> ). 2 Then there is a constant C1 (N ) such that for y + ξ ∈ N ,
(3.2.5)
(3.2.6)
|B(y, ξ)| + |Dy B(y, ξ)| + |ξ||Dξ B(y, ξ)| ≤ C1 |ξ|3 .
This holds because, locally, N is the graph of second fundamental form A over Ty N ; that is, for y + ξ ∈ N , 1 (3.2.7) y + ξ = y + ξy> + A(y)(ξy> , ξy> ) + O(|ξ|3 ). 2 For ξ, η ∈ Rp and x ∈ Ω0 , define (3.2.8)
> > Qu (x, ξ, η) = A(u(x))(ξu(x) , ηu(x) ) · A(u(x))(∇u(x), ∇u(x)).
(3.2.9)
Ru (x, ξ) = 2B(u(x), ξ) · A(u(x))(∇u(x), ∇u(x)).
By the definition (3.2.2) of Su , (3.2.5), (3.2.8)-(3.2.9) and the fact that A(u)⊥Tu N , we verify Su (x, ξ) = Qu (x, ξ, ξ) + Ru (x, ξ) = 2ξ ⊥ · A(u)(∇u, ∇u) = 2ξ · A(u)(∇u, ∇u). This combined with (3.2.4) gives (3.2.1). The (3.2.3) follows from the definition (3.2.9) of Ru , (3.2.6) and the fact |∇u| ≤ kuk1;0 /ρ. 9
3.3. Remark. For ξ = H01 (Ω, Tu N ), if we let v = ut : (ε, ε) → H 1 (Ω, N ) be a curve with u0 = u and d dt t=0 ut = ξ in (3.2.1), then (3.2.3) implies that (3.3.1)
d2 E(ut ) = 2 dt2 t=0
Z |∇ξ|2 − Qu (x, ξ, ξ). Ω
In particular, if N = Sp−1 , we can take Ru (x, ξ) = 0 and Su (x, ξ) = Qu (x, ξ, ξ) = |ξ|2 |∇u|2 ; therefore, we have for ξ = v − u, Z (3.3.2) E(v) − E(u) = |∇ξ|2 − |∇u|2 |ξ|2 . Bn
Indeed, in this case, because 2ξ · u + |ξ|2 = 0, A(u)(ξ, η) = −(ξ · η)u, we have, in (3.2.4) 2ξ · A(u)(∇u, ∇u) = −2ξ · u|∇u|2 = |ξ|2 |∇u|2 (If we apply theorem 3.2, we will have B(u, ξ) = −(ξ · u)2 u/2, Qu (x, ξ, ξ) = |∇u|2 (|ξ|2 − (ξ · u)2 ) and Ru (x, ξ) = |∇u|2 (ξ · u)2 .) Thus in the case N = Sp−1 , both (3.1.2) and (3.1.3) can be written as Z
Z |∇ξ|2 − |∇u|2 |ξ|2 ≥ µ
(3.3.3) Bn
Ω
|ξ|2 ρ2
for all ξ ∈ H01 (Ω, Rp ) that satisfy |ξ + u| = 1 and ξ ∈ H01 (Ω, Tu N ), respectively. 2 m+1 1 Similarly, for N = Sm : |u|2 + |z| a = {(u, z) ∈ R a2 = 1}, with a > 0, if (u, 0) ∈ H (Ω, N ) is an equator p 1 1 map, then for (ξ, z) ∈ H0 (Ω, R ) (p = m + 1) satisfying (u + ξ, z) ∈ H (Ω, N ), we have Z Z u (3.3.4) E((u + ξ, z)) − E((u, 0)) = |∇ξ|2 − |∇u|2 |ξ|2 + |∇z|2 − |∇ |2 |z|2 . a Bn Bn 3.4. Theorem. A harmonic map u ∈ H 1 (Ω, N ) is strictly stable if and only if u is strictly nearby minimizing. Proof : Suppose that u is strictly minimizing with rate µ > 0. If ξ ∈ H01 (Ω, Tu N ) ∩ L∞ , then ut = π(u + tξ) d satisfies dt u = ξ and |ut − u| ≤ C|t||ξ|. So when t is sufficiently small, by strict nearby minimality of u, t=0 t Z E(ut ) − E(u) ≥ µ Ω
Thus
|ut − u|2 . ρ2
d2 E(ut ) − E(u) E(ut ) = 2 lim ≥ 2µ t→0 dt2 t=0 t2
Z Ω
|ξ|2 . ρ2
This shows the necessity. To show sufficiency, suppose that u ∈ C 1;0 (Ω0 , N ) is strictly stable with rate µ > 0. Let (3.4.1)
ξ ∈ H01 (Ω, Rp ),
ξ + u ∈ N,
|ξ| ≤ δ,
with δ > 0 being chosen. Define η = ξu> and χ = ξu⊥ = ξ − η. Then η ∈ H01 (Ω, Tu N ) and by strict stability of u and formula (3.3.1) Z
Z |∇η|2 − Qu (x, η, η) ≥ µ
(3.4.2) Ω
Ω
10
|η|2 . ρ2
From the definition of C 1;0 (Ω0 , N ) and the fact N ∈ C 4 , we have (3.4.3)
|∇u(x)| ≤
(3.4.4)
|ξu> | ≤ |ξ|,
kuk1;0 , ρ(x)
kAkC 2 (N ) ≤ C2 . |ξ| > |∇ξu | ≤ C3 |∇ξ| + . ρ
Here and in the following, unless otherwise indicated, the constants Ci depend only on N , kuk1;0 , m and p. Now from (3.2.5), we have 1 χ = ξu⊥ = B(x, ξ) + A(u)(ξu> , ξu> ). 2 Using (3.2.6), (3.4.3) and (3.4.4), we get (3.4.5)
|χ| ≤ C4 [|ξ|3 + |ξu> |2 ] ≤ 2C4 |ξ|2 ≤ 2C4 δ|ξ|.
(3.4.6)
|∇χ| ≤ |Du B(u, ξ)||∇u| + |Dξ B(u, ξ)||∇ξ|+ 1 + |Du A(u)(∇u, ξu> , ξu> )| + |A(u)(∇ξu> , ξu> )| 2 3 |ξ| |ξu> |2 |ξ|2 2 ≤ C5 + |∇ξ||ξ| + + |∇ξkξ| + ρ ρ ρ |ξ| ≤ C6 δ |∇ξ| + . ρ
By Schwartz’s inequality, (3.4.5) and (3.4.6), we have the following |ξ|2 (3.4.7) |∇χ|2 ≤ 2C62 δ 2 |∇ξ|2 + 2 . ρ (1 − C7 δ)|ξ|2 ≤ |η|2 ≤ (1 + C7 δ)|ξ|2 . |ξ|2 |∇η|2 ≤ (1 + C8 δ)|∇ξ|2 + C8 δ 2 . ρ
(3.4.8) (3.4.9)
Since Q is bilinear and η = ξ − χ, (3.4.10)
Qu (x, η, η) = Qu (x, ξ, ξ) − 2Qu (x, ξ, χ) + Qu (x, χ, χ).
By definition of Q, (3.4.3) and (3.4.5), |Qu (x, ξ, χ)| = |A(u)(ξu> , χ> u ) · A(u)(∇u, ∇u)| |ξ|2 ≤ C9 |ξ||χ||∇u|2 ≤ C10 δ 2 . ρ
(3.4.11)
(3.4.12)
(3.4.13)
> |Qu (x, χ, χ)| = |A(u)(χ> u , χu ) · A(u)(∇u, ∇u)| |ξ|2 ≤ C11 |χ|2 |∇u|2 ≤ C12 δ 2 2 . ρ
|Qu (x, ξ, ξ)| ≤ |A(u)(ξu> , ξu> ) · A(u)(∇u, ∇u)| ≤ C13
Thus (3.4.10)-(3.4.12) imply (3.4.14)
Q(x, η, η) ≤ Q(x, ξ, ξ) + C14 δ 11
|ξ|2 , ρ2
|ξ|2 . ρ2
and (3.4.13), (3.4.14) imply (3.4.15)
Q(x, η, η) ≤ (1 + C8 δ)Q(x, ξ, ξ) + C15 δ
|ξ|2 . ρ2
From (3.2.3), (3.4.16)
|Ru (x, ξ)| ≤ C0 δ
|ξ|2 . ρ2
Putting (3.4.9), (3.4.15) and (3.4.8) into (3.4.2), we get Z
|ξ|2 |ξ|2 (1 + C8 δ)|∇ξ| + C8 δ 2 − (1 + C8 δ)Q(x, ξ, ξ) + C15 δ 2 ≥ ρ ρ Ω
Z
2
µ(1 − C7 δ) Ω
|ξ|2 . ρ2
Combining this with the definition of Su (x, ξ) and (3.4.16), we get Z (1 + C8 δ)
Z |ξ|2 |∇ξ|2 − Su (x, ξ) ≥ (µ − C16 δ) . 2 Ω Ω ρ
Choose δ small so that µ1 = (µ − C16 δ)/(1 + C8 δ) > 0, then Z
Z |∇ξ|2 − Su (x, ξ) ≥ µ1 Ω
Ω
|ξ|2 , ρ2
for all ξ satisfying (3.4.1). That is u is strictly δ-nearby minimizing with rate µ1 . 3.5. Theorem. Suppose that u ∈ C 1;0 (Ω0 , N ) is a strictly minimizing harmonic map. Then there is an ε > 0 depending only on kuk1;0 , m and p, such that any harmonic map v ∈ C 1;0 (Ω0 , N ) with kv − uk1;0 ≤ ε is also strictly minimizing. Proof : Take δ = δ(N ) > 0 such that for any x ∈ N4δ ≡ {x ∈ Rp : dist(x, N ) ≤ 4δ}, there exists unique π(x) ∈ N such that dist(x, N ) = |x − π(x)|. Note that π ∈ C 3 , since N ∈ C 4 . Suppose that u ∈ C 1;0 (Ω0 , N ) is harmonic and strictly minimizing with rate µ > 0 and v ∈ C 1;0 (Ω0 , N ) is harmonic satisfying (3.5.1)
kv − uk1;0 ≤ ε
with ε ∈ (0, δ). We show that v also strictly minimizing if ε is small enough. Suppose that η ∈ H01 (Ω, Rp ) and η + v ∈ N . Define ξ = π(u + η) − u, then ξ is well defined because dist(u + η, N ) ≤ kv − uk1;0 ≤ ε ≤ δ. Noticing that ξ ∈ H01 (Ω, Rp ) and ξ + u ∈ N , by strict minimality of u, we have Z Z |ξ|2 . (3.5.2) |∇ξ|2 − Su (x, ξ) ≥ µ 2 Ω Ω ρ Let χ = ξ − η = π(u + η) − (u + η). To estimate χ we consider the map (3.5.3)
F : Q2δ ≡ {(y, z) ∈ N2δ × Rp , y + z ∈ N2δ } → Rp F (y, z) = π(y + z) − (y + z). 12
Since dist(∂Q2δ , Qδ ) ≥ δ and π ∈ C 3 , we can extend F to Rp × Rp → Rp so that for some constant C(δ, N ), |Dyα Dzβ F | ≤ C, |α| + |β| ≤ 3. Now since F (v, η) = 0, we have Z
1
F (u, η) =
Dy F (v + t(u − v), η)(u − v) dt. 0
Using the fact that F (u, 0) = 0 in this equality, we get Z 1 (3.5.4) F (u, η) = [Dy F (v + t(u − v), η) − Dy F (v + t(u − v), 0)](u − v) dt 0
Z
1
Z
1
Dy Dz F (v + t(u − v), sη)[(u − v) ⊗ η] dtds.
= 0
0
From (3.5.4), (3.5.1) and by using Schwartz’s inequality, (3.5.5)
|χ| = |F (u, η)| ≤ |Dy Dz F | |u − v||η| ≤ C2 ε|η|.
(3.5.6)
(1 − C3 δ)|η|2 ≤ |ξ|2 ≤ (1 + C3 δ)|η|2 .
(3.5.7)
(3.5.8) (3.5.9)
|∇χ| ≤ C4 |∇v| + |∇u| + |η| |u − v||η|+ + C4 |∇u − ∇v||η| + |u − vk∇η| |η| . ≤ C5 ε |∇η| + ρ 2 |η| 2 |∇χ|2 ≤ 2C52 ε2 + |∇η| . ρ2 |η|2 |∇ξ|2 ≤ (1 + C6 ε)|∇η|2 + C6 ε 2 . ρ
Next we need to consider the maps ut = π(u + t(v − u)), t ∈ [0, 1]. As in (3.4.3) and (3.4.4) we have (3.5.10) (3.5.11)
|∇ut (x)| ≤ C7
kuk1;0 + δ , ρ(x)
|ηu>t | ≤ |η|,
kAkC 2 (N ) ≤ C7 , kπkC 3 (N2δ ) ≤ C7 , |η| |∇ηu>t | ≤ C8 |∇η| + . ρ
By definition (3.2.2) of Su and the relation ξ = η + χ, (3.5.12)
Su (x, ξ) = Qu (x, ξ, ξ) + Ru (x, ξ) = Qu (x, η, η) + 2Qu (x, η, χ) + Qu (x, χ, χ) + Ru (x, ξ) = Sv (x, η) + [Qu (x, η, η) − Qv (x, η, η)]+ [Ru (x, ξ) − Rv (x, η)] + 2Qu (x, η, χ) + Qu (x, χ, χ).
By definition (3.2.8) of Q, (3.5.10) and (3.5.11), we get (3.5.13)
|Qu (x, η, η) − Qv (x, η, η)| = |A(ut )(ηu>t , ηu>t ) · A(ut )(∇ut , ∇ut )|t=1 t=0 | Z 1 d = [A(ut )(ηu>t , ηu>t ) · A(ut )(∇ut , ∇ut )]dt dt 0
≤ C9 ε
|η|2 . ρ2 13
Similarly, by definition (3.2,9) of R, (3.5.10)-(3.5.11) and (3.5.5)-(3.5.6), (3.5.14)
|Ru (x, ξ) − Rv (x, η)| ≤ |Ru (x, ξ) − Rv (x, ξ)| + |Rv (x, ξ) − Rv (x, η)| |η|2 |η|2 + |ξ|2 |η|2 ≤ C10 |v − u| 2 + C10 |η − ξ| ≤ C ε . 11 ρ ρ2 ρ2
By definition (3.2.8) of Q and (3.5.5) (similar to (3.4.11)-(3.4.12)), (3.5.15)
|Qu (x, η, χ)| ≤ C12 |η||χ||∇u|2 ≤ C13 ε |Qu (x,χ, χ)| ≤14 Cε2
(3.5.16)
|η|2 . ρ2
|η|2 . ρ2
Putting (3.5.13)-(3.5.16) into (3.5.12), we get (3.5.17)
−Su (x, ξ) ≤ −Sv (x, η) + C15 ε
|η|2 . ρ2
To write (3.5.17) into the needed form, we note that (3.5.18)
|Sv (x, η)| ≤ |Qv (x, η, η)| + |Rv (x, η)| ≤ C16
|η|2 , ρ2
thus (3.5.17) together with (3.5.18) gives (3.5.19)
−Su (x, ξ) ≤ −(1 + C6 ε)Sv (x, η) + C17 ε
|η|2 ρ2
Substituting (3.5.9), (3.5.19) and (3.5.6) into (3.5.2), we get Z
Z 2
(1 + C6 ε)
|∇η| − Sv (x, η) ≥ µ(1 − C18 ε) Ω
Ω
|η|2 . ρ2
Choose ε small so that µ1 = (µ − C18 ε))/(1 + C6 ε) > 0, then Z
Z |∇η|2 − Sv (x, η) ≥ µ1
Ω
Ω
|η|2 , ρ2
for all η ∈ H01 (Ω, Rp ) satisfying η + v ∈ N . This shows that v is strictly minimizing with rate µ1 , if kv − uk1;0 ≤ ε. Remark 3.6. From the proof of Theorem 3.5, we see that ε and µ1 depend only on kuk1;0 , µ, m and p; furthermore we have µ1 → µ when ε → 0+. Thus we have the following Corollary 3.7. If ui ∈ C 1;0 (Ω0 , N ) is a sequence of strictly minimizing harmonic maps with rates µi ≥ µ > 0 and ui → u in C 1;0 (Ω0 , N ), then u is also a strictly minimizing harmonic map with rate µ. Proof : Since ui → u in C 1;0 (Ω0 , N ), we have ui → u in H 1 (Ω, N ) and kui k1;0 ≤ K for some K ∈ (0, ∞). So u is harmonic. By Remark 3.6, there are an ε = ε(K, µ, m, p) > 0 and µ1 = µ1 (K, µ, m, p) > 0 such that if v ∈ C 1;0 (Ω0 , N ) is harmonic and kui − vk1;0 ≤ ε for some i, then v is strictly minimizing with rate µ1 . In particular, when i large, we have kui − uk1;0 ≤ ε; so u is strictly minimizing with rate µ1 . Note that µ1 can be arbitrarily close to µ if ε is close to 0. Therefore u is actually strictly minimizing with rate µ. 14
By the same technique in Theorem 3.5, we can show that strict stability and strict nearby minimality are also preserved under small perturbation. We have the following theorem, whose proof is omitted. 3.8. Theorem. If u ∈ C 1;0 (Ω0 , N ) is strictly minimizing (strictly stable, or strictly nearby minimizing) harmonic map with rate µ > 0, then there are positive constants ε and µ1 depending on kuk1;0 , µ, N , m, p such that any harmonic map v ∈ C 1;0 (Ω0 , N ) with kv − uk1;0 ≤ ε is strictly minimizing (strictly stable, or strictly nearby minimizing) with rate µ1 . If ui ∈ C 1;0 (Ω0 , N ) is a sequence of strictly minimizing (strictly stable, or strictly nearby minimizing) harmonic maps with rates µi ≥ µ > 0 and ui → u in C 1;0 (Ω0 , N ), then u is a also strictly minimizing (strictly stable, or strictly nearby minimizing) harmonic map with rate µ.
§ 4. Structure of the Space of Harmonic Maps 4.1. Here we study the structure of the space of harmonic maps in C k,α;0 (Bm 1 \ {0}, N ) near a homogeneous harmonic map that is singular only at the origin. We mainly prove that this space is Banach manifold (Theorem 4.8) and give precise estimates on the Fredholm indices of the Jacobi operator (Theorem 4.4). N. Smale has recently proved the Fredholm property of “conic” operators that include the Jacobi operator of a harmonic map with isolated singularities and unique tangent maps [SN3]. m Let B = Bm 1 (0) and Z = {0} and denote B0 = B1 (0)\{0}. Suppose that u : B0 → N is a homogeneous x harmonic with ϕ = u|∂B ∈ C 2,α (Sm−1 , N ). Then we have that u(x) = ϕ( |x| ) and ϕ is also harmonic. Let Jϕ and Ju be the Jacobi operators with respect to ϕ and u repectively. Using the polar coordinates x = rθ (r = |x|, θ ∈ Sm−1 ) on B, we can write, by (1.7.2), (4.1.1)
Jϕ η = ∆S η − 2A(ϕ)(∇S ϕ, ∇S η) − DA(ϕ)(η, ∇S ϕ, ∇S ϕ)
and (4.1.2)
Ju κ =
∂2 m−1 ∂ 1 κ+ κ + 2 Jϕ κ(r·) ∂r2 r ∂r r
for η ∈ C 2,α (Sm−1 , Tϕ N ) and κ ∈ C 2,α;ν (B0 , Tu N ), where ∇S is the covariant differentiation on Sm−1 . Let µ1 ≤ µ2 ≤ · · · be the eigenvalues of Jϕ and η1 , η2 , ..., be the corresponding orthonormal (with respect to h·, ·iL2 (Sm−1 ) ) eigenmaps in C 2,α (Sm−1 , Tϕ N ). The following lemma gives a relation between the first eigenvalue µ1 of Jϕ and the first eigenvalue λ of Ju . 4.2. Lemma. λ≥
(m − 2)2 + µ1 . 4
Proof : Argue as in the proof of [CHS]. 4.3. Now for the Jacobi operator Ju : C 2,α;ν (B0 , Tu N ) → C 0,α;ν−2 (B0 , Tu N ), we want to solve Ju κ = f for κ ∈ C 2,α;ν (B0 , Tu N ) with f ∈ C 0,α;ν−2 (B0 , Tu N ). Writing (4.3.1)
κ(rθ) =
∞ X
κi (r)ηi (θ),
i=1
f (rθ) =
∞ X i=1
15
fi (r)ηi (θ)
with κi (r) = hκ(r·), ηi iL2 (Sm−1 ) , fi (r) = hf (r·), ηi iL2 (Sm−1 ) , then formally, Ju κ = f is equivalent to κ00i (r) +
(4.3.2)
m−1 0 1 κi − 2 µi = fi (r), r r
i = 1, 2, 3, ...,
which are nonhomogeneous Euler equations. Let for i = 1, 2, 3, ..., 2−m γi = + 2
r
(m − 2)2 + µi , 4
γi−
r
2−m = − 2
(m − 2)2 + µi . 4
Then a general solution gi of the homogeneous equation associated with (4.3.2) is
(4.3.3)
2 γi γi }; i ∈ I1 ≡ {i : µi < − (m−2) ai Rer + bi Imr , 4 2 2−m 2−m gi (r) = ai r 2 + bi r 2 log r, i ∈ I2 ≡ {i : µi = − (m−2) }; 4 (m−2)2 γi γi− ai r + bi r , i ∈ I3 ≡ {i : µi > − 4 },
with arbitrary constants ai and bi . Note that γi = γ¯i− are complex for i ∈ I1 , γi = γi− = γi− < 2−m < γi for i ∈ I3 . A particular solution Fi (r) of (4.3.2) is 2 (4.3.4)
Fi (r) =
2−m 2
for i ∈ I2 and
Rr Rτ Re rγi 0 τ −m+1−2γi 0 sm−1+γi fi (s)dsdτ, γi < ν; R R r τ Re rγi 1 τ −m+1−2γi 0 sm−1+γi fi (s)dsdτ, γi ≥ ν.
Note that Fi (r) satisfies that |Fi (r)| ≤ Crν for i such that γi 6= ν. Thus (4.3.2) has a general solution κi (r) = gi (r) + Fi (r). Let ∞ ∞ X X G(rθ) = gi ηi (θ), F(f )(rθ) = Fi (r)ηi (θ), i=1
i=1
then the solution κ of Ju κ = f can be written κ(rθ) = G(rθ) + F(f )(rθ). Denote by Kν0 (Ju ) and Kν (Ju ) be the kernels of Ju in C02,α;ν (B0 , Tu N ) and C 2,α;ν (B0 , Tu N ), respectively, and Rν (Ju ) = Ju (C02,α;ν (B0 , Tu N )). 4.4. Theorem. (a). If ν 6= γj and ν >
2−m 2 ,
then
Kν (Ju ) = {κ ∈ C 2,α;ν (B0 , Tu N ) : κ(rθ) =
X
ai rγi ηi (θ)}.
γi >ν
(b). If ν 6= γj , then #{i : ν ≤ Re γi− }, dim(Kν0 (Ju )) = #{i : γi is complex}, 0,
if ν < if ν = if ν >
and codim(Rν (Ju )) = #{i : γi < ν}.
16
2−m 2 ; 2−m 2 ; 2−m 2 .
Proof : Suppose κ ∈ C 2,α;ν (B0 , Tu N ) satisfies Ju κ = 0, then κ(rθ) = G(rθ) with |κ(r·)|L2 (Sm−1 ) =
∞ X
gi2 (r),
i=1
where gi are given in (4.3.3). Since ν >
2−m 2 ,
(4.4.1)
κ(rθ) =
X
ai rγi ηi (θ).
γi >ν
Conversely, any κ ∈ C 2,α;ν (B0 , Tu N ) of this form is in Kν (Ju ). This shows (a). 2,α;ν To show (b) when ν < 2−m (B0 , Tu N ), then from (4.4.1), we 2 , suppose that Ju κ = 0 and κ ∈ C0 must have |gi (r)| = O(rν ) and gi (1) = 0. Thus κ has the following expansion κ(rθ) =
X
bi Imrγi +
i∈I1
X
bi r
2−m 2
X
log r +
i∈I2
−
ai (rγi − rγi ).
i∈I3 ∩{γi− >ν}
Thus dim(Kν (Ju )) = #{i : ν ≤ Re γi− }. The other cases ν ≥ 2−m are similar. 2 Now we estimate codim(Rν (Ju )). First we show that F(f ) is a solution of Ju κ = f in C 2,α;ν (B0 , Tu N ) for f ∈ C 0,α;ν−2 (B0 , Tu N ). Write (4.4.2)
F(f ) =
∞ X i=1
Fi (r)ηi (θ) =
X
Fi (r)ηi (θ) +
γi 0 such that any harmonic map v ∈ V = {v ∈ C 2,α;0 (B0 , N ) : kv − uk2,α;ν < } is obtained in this way. (c). U and can be chosen so that V ∩ H is a smooth manifold C 2 diffeomorphic to U . The tangent space Tu H = Kν (Ju ). In particular, all Jacobi fields are integrable. Proof : Any ψ ∈ C22,α (Sm−1 , Tu N ) can be extended to a Jacobi field ψ¯ ∈ Kν (Ju ). P ψ(θ) = i∈I2 ai ηi (θ), then ¯ ψ(rθ) =
(4.7.1)
X
ai rγi ηi (θ).
i∈I2
(See Theorem 4.4). Consider the map (4.7.2)
Ψ : C22,α (Sm−1 , Tu N ) × C∗2,α;ν (B0 , Tu N ) → C 0,α;ν−2 (B0 , Tu N ) 18
That is, if
defined by Ψ(ψ, κ) = Pu ◦ H(π(u + ψ¯ + κ)). We note the following fact: if kv − uk2,α;ν is small, then (4.7.3)
H(v) = 0 ⇐⇒ Pu ◦ H(v) = 0.
Indeed, if Pu ◦ H(v) = 0, then hH(v), H(v)iL2 = hH(v), H(v) − Pu ◦ H(v)iL2 = hH(v), [Dπ(v) − Dπ(u)] ◦ H(v)iL2 ≤ kDπ(v) − Dπ(u)k hH(v), H(v)iL2 . So hH(v), H(v)iL2 = 0 when v is close to u so that kDπ(v) − Dπ(u)k < 1. Note that D2 Ψ(0, 0)κ = Pu ◦ DH(u)κ = Ju κ. By Theorem 4.6, Ju is an isomorphism. By the implicit function theorem, there are neighborhoods U of 0 in C22,α (Sm−1 , Tu N ) and W of 0 in 2,α;ν C∗ (B0 , Tu N ) and a smooth map Q : U → W such that for any ψ ∈ U , Q(ψ) is the unique solution of (4.7.4)
Ψ(ψ, Q(ψ)) = Pu ◦ H(π(u + ψ¯ + Q(ψ))) = 0.
Let F (ψ) = π(u + ψ¯ + Q(ψ)). Then F is a smooth map from U into some neighborhood V of u in C 2,α;0 (B0 , N ). By (4.7.4) and (4.7.3), F (ψ) is harmonic. To show that DF (0) is an immersion, first note that for ξ ∈ C22,α (Sm−1 , Tu N ), ξ∗ ≡ DF (0)ξ ∈ C 2,α;ν (B0 , Tu N ), and Π2 (ξ∗ − ξ) = 0 on Sm−1 . Indeed this follows from the fact that ξ∗ = DF (0)ξ = ξ¯ + DQ(0)ξ, where DQ(0)ξ ∈ C∗2,α;ν (B0 , Tu N ). For ξ ∈ C 2,α;ν (B0 , Tu N ), define ξ∗ = (Π2 ξ|Sm−1 )∗ and Λ ξ = (Π2 ξ|Sm−1 , ξ − ξ∗ ). Then one readily sees that Λ C 2,α;ν (B0 , Tu N ) −→ C22,α (Sm−1 , Tu N ) × C∗2,α;ν (B0 , Tu N ) is an isomorphism. Consider the following sequence DF (0)
C22,α (Sm−1 , Tu N ) −→ C 2,α;ν (B0 , Tu N ) Pr
Λ
1 −→ C22,α (Sm−1 , Tu N ) × C∗2,α;ν (B0 , Tu N ) −→ C22,α (Sm−1 , Tu N )
where P r1 is the projection. It is easy to see that this gives an identity. So DF (0) is an immersion [L, II, 2]. Now we show that if is small and v ∈ V = {v : kv − uk2,α;ν < } is a harmonic map, then v = F (ψ) with ψ = Π2 π −1 v|Sm−1 . Since π is a locally a diffeomorphism between Tu N and N (near u ∈ N ), we have that for each x ∈ B0 , there exists a unique λ(x) ∈ Tu(x) N such that v(x) = π(u(x) + λ(x)), ¯ where ψ¯ is defined by and λ, denoted by λ = π −1 v, smoothly depends on v. Let ψ = Π2 λ and κ = λ − ψ, 2,α;ν (4.7.1). Then κ ∈ C∗ (B0 , Tu N ) and v = π(u + ψ¯ + κ). That v is harmonic implies that Ψ(ψ, κ) = Pu ◦ H(π(u + ψ¯ + κ)) = 0. 19
By the uniqueness of Q for small ψ, we get κ = Q(ψ) and then v = F (ψ). That V ∩ H is a manifold follows since DF (0) is an immersion [L, II,2]. To prove that Tu H = Kν (Ju ) we need only to show that if κ ∈ C 2,α;ν (B0 , Tu N ) is a Jacobi field, then there is a one parameter family d of harmonic maps ut ∈ H with u0 = u and dt u = κ. Indeed since κ|Sm−1 ∈ C22,α (Sm−1 , Tu N ) (by t=0 t Theorem 4.4(a)), by (a), ut = F (tκ) is a family of harmonic maps (for all small t) and we have d d ut = t=0 F (tκ) = κ + DQ(0)κ t=0 dt dt (note that κ ¯ = κ). In fact DQ(0)κ = 0 because DQ(0)κ is a Jacobi field in C∗2,α;ν (B0 , Tu N ) on which Ju is d one-to-one. So dt u = κ. t=0 t
§ 5. Applications to Some Examples of Homogeneous Harmonic Maps In this section, we apply Theorem 4.7 and Theorem 3.8 to some well-known examples of harmonic maps. We first examine the stability and minimality of those maps, some of which are well-known, while some others are new. ∂u m−1 5.1. Lemma. If u ∈ H 1 (B, N ) and either u is a stationary harmonic map with |S ∈ L2 (Sm−1 ), ∂n 1 (this holds, e.g., if u ∈ H 2,2 (B, N )), or u ∈ Cloc (B0 , N ) is a harmonic map, then Z Z Z ∂u (5.1.1) |∇tan u|2 = | |2 + (m − 2) |∇u|2 . Sm−1 Sm−1 ∂n B Proof : When u is a stationary harmonic map in H 2,2 (B, N ), this was shown, for example, in [H]. Also one can show (5.1.1) by modifying the proof of the monotonicity identity in [HL] as follows. Starting with the last identity on page 570 in [HL] (for p = 2, r = 1), first take R = 1 and s → 1, then let r ↑ s = 1 (instead ∂u m−1 of s ↓ r). In this process, only that |S ∈ L2 (Sm−1 ) is needed. ∂n 5.2. Lemma. Suppose that u ∈ C 1;0 (B0 , N ) is a homogeneous harmonic map and v ∈ H 1 (B, N ) belongs to either of two classes in Lemma 5.1 which satisfies v = u on Sm−1 but v 6= v. Then E(v) < E(u) In particular, u is the unique energy minimizer (or unique harmonic map) as long as u has least energy (among the harmonic maps). Proof : By the assumption and Lemma 5.1 we have that (5.1.1) holds for v Z Z Z ∂v (5.2.1) |∇tan v|2 = | |2 + (m − 2) |∇v|2 . Sm−1 Sm−1 ∂n B ∂v ∂u ∂v 6= 0, for otherwise we would have v = u and = ∂n ∂n ∂n on Sm−1 . By unique continuation theorem [ML2, Prop. 3.2], we have v ≡ u, a contradiction to that v 6= u. R ∂v Thus we have Sm−1 | |2 > 0, which impies that ∂n Z Z Z Z 1 1 (5.2.2) |∇v|2 < |∇tan v|2 = |∇tan u|2 = |∇u|2 . m − 2 Sm−1 m − 2 Sm−1 B B Now note that since v 6= u, we must have that
This implies that if u and v both have least energy (or least energy among harmonic maps), then v ≡ u.
20
5.3. Proposition. For any harmonic map ω : S2 → S2 , let x 3 2 u : B → S , u(x) = ω , x ∈ B3 \ {0}, |x| be the corresponding homogeneous harmonic map. Then u ∈ C 2,α;0 (B30 , S2 ) for any 0 < α < 1. Furthermore, we have (a). ω is stable. u is strictly stable with rate 14 . (b). ω is energy minimizing among the maps having same degree as ω has. u is the unique energy minimizing among the maps in C 1;0 (B30 , S2 ) having same boundary data. Proof : It is a well-known result in [BCL] that ω has least energy among the maps from S2 to itself having same degree. Consequently, ω is stable, since the continuity of degree implies that for any smooth variation d ωt of ω, deg(ωt ) = deg(ω) when t is small, then E(ut ) ≥ E(u) and so dt E(ut ) ≥ 0. t=0 1 It follows from Lemma 4.2, u is strictly stable with rate 4 . To show that u is strictly minimizing, let v ∈ C 1;0 (B30 , S2 ) with v|S2 = ω. Again by continuity of degree, deg(v(r·)) = deg(ω) for each r > 0, thus by minimality of ω, Z Z 1 Z Z ∂v |∇v|2 = r |∇S2 v(r·)|2 dS2 dr + | |2 3 2 3 B 0 S B ∂r Z 1 Z Z ∂(v − u) 2 ≥ r |∇S2 ω|2 dS2 dr + | | ∂r 2 3 0 S B Z Z 1 1 v−u 2 ≥ |∇S2 ω|2 dS2 + | | 2 2 4 B3 r Z Z S v−u 2 1 | | . = |∇u|2 + 4 B3 r B3 5.4. Proposition. Let I : Sm−1 → Sm−1 (m ≥ 4) be the identity and u : Bm → Sm−1 be the homogeneous x extension, that is, u(x) = |x| . Then (a). I is not stable for all m ≥ 4. In fact, index(I) = m. u : Bm → Sm−1 is stable for all m = 3 and strictly stable for m = 3 and all m ≥ 5. (b). u is the unique minimizer in H 1 (Bm , Sm−1 ) for all m ≥ 3 and strictly minimizing for m ≥ 7. Proof : That index(I) = m is computed by R. T. Smith [SRT]. That u is the unique minimizer in H 1 (Bm , Sm−1 ) was proved in [LF] and [JK] and [BCL]. The uniqueness can also be seen from Lemma 5.2 above. The stability and strict stability was shown in [BA]. Now we review the harmonic equator maps from the unit ball Bm to the spheroid Sm a = {(u, z) ∈ |z|2 m+1 2 R : |u| + a2 = 1}, with a > 0. x 5.5. Proposition.Let u∗ : Bm → Sm a , ∗ u(x) = ( |x| , 0), be the equator map. Then
(a). u∗ is stable for a ≥
√ 2 m−1 m−2
and and m ≥ 3;
u is strictly stable for m = 3 or m ≥ 5 and a > √ 2 m−1 m−2 , then u∗ is unstable. √ m−1 a > 2 m−2 and m ≥ 3, then u∗ is √ m−1 m ≥ 7 and a > 2 m−2 , then u∗ is
√ 2 m−1 m−2 .
If a < (b). If If
strictly minimizing in rotationally symmetric maps. strictly minimizing in H 1 (B, Sm a ).
Proof : These properties were proved in [BA] and [MS]. The strictly stability also follows from Remark 3.3 and the Lemma 4.2. 21
Suppose u ∈ C 1;0 (B0 , N ) is a homogeneous harmonic map with ω = u|Sm−1 q . Let L = index(ω) + 2 2−m nullity(ω) + 1, then µL is the first positive eigenvalue of Jω and γL = 2 + (m−2) + µL > 0. Let 4
ν ∈ (0, γL ) and 0 < α < 1. Recall that C22,α (Sm−1 , Tu N ) is the subspace of C 2,α (Sm−1 , Tu N ) spaned by the eigenvectors of ω corresponding to positive eigenvalues. Combining the Theorem 4.7 and Theorem 3.8, we have the following 5.6 Theorem. There is a neighborhood U of 0 in C22,α (Sm−1 , Tu N ) and a small number > 0 so that, for each ϕ ∈ U , there exists a harmonic map v ∈ V = {v ∈ C 2,α;0 (B0 , N ) : kv − uk2,α;ν < } so that v|Sm−1 = π(ω + ϕ). We can choose U and to be small, so that If u is strictly stable, then v is also strictly stable. Thus the harmonic maps in V form a manifold. If u strictly minimizing, then v is also strictly minimizing. Therefore, the harmonic maps in V also form a manifold. 5.7. Remark. Note that strict minimality implies uniqueness, thus the strict minimality of ω implies the uniqueness of those v with v|Sm−1 close to ω. 5.8. Remark. Proposition 5.6 applies especially to those maps in Propositions 5.3, 5.4 and 5.5. § 6. Global Structure of Smooth Harmonic Maps and Applications In this section we consider only smooth harmonic maps and obtain various results by modifying arguments of Brian White’s treatment [WB] of minimal immersions. Theorem 6.3 shows that any Jacobi field in C k,α (Ω, Tu N ) is integrable. Theorems 6.2 and 6.4 describe the local and global structures of the space H of the smooth harmonic maps. 6.1. Proposition. Suppose that Ju : C k,α (Ω, Tu N ) → C k−2,α (Ω, Tu N ) is the Jacobi operator with respect to a harmonic map u ∈ C k,α (Ω, N ). Let K0 be the kernel of Ju in C0k,α (Ω, Tu N ) and [K0 ]⊥ is orthogonal complement in L2 (Ω, Tu N ). Then dim (K0 ) < ∞,
Im Ju = [K0 ]⊥ ∩ C k−2,α (Ω, Tu N ).
So Ju is Fredholm of index 0. Proof. Being considered as an operator defined on H01 (Ω, Tu N ), Ju is self-adjoint (by (1.7.1)) and uniformly elliptic. Furthermore by standard elliptic theory any solution κ of Ju κ = f for f ∈ C k−2,α (Ω, Tu N ) is in C k,α (Ω, Tu N ) and h i kκkk,α ≤ C kIκkk−2,α + kJu κkk−2,α , where I : C0k,α (Ω, Tu N ) → C k−2,α (Ω, Tu N ) is the inclusion, which is compact. This inequality implies that Ju has closed range, by [T, Proposition 3.1]. Thus K0 is equal to the kernel of Ju in H01 (Ω, Tu N ) which is finitely dimensional. By Fredholm alternative, Ju κ = f is solvable iff f ∈ K0⊥ . So Im Ju = [K0 ]⊥ ∩ C k−2,α (Ω, Tu N ).
6.2. Theorem (Local Structure). Suppose u ∈ H and K0 = K0 (Ju ). Then there exist smooth maps F : C k,α (∂Ω, N ) × K0 → C k,α (Ω, N ), g : C k,α (∂Ω, N ) × K0 → K0 , 22
defined on some neighborhood U = U1 × U2 of (ϕ, 0), ϕ = u|∂Ω, and satisfying the following: (a). F (u|∂Ω, 0) = u. F (ψ, κ)|∂Ω = ψ for all (ψ, κ) ∈ U . F (ψ, κ) is harmonic if and only if g(ψ, κ) = 0. (b). D2 F (ϕ, 0)|K0 = id. DF (ϕ, 0) : C k,α (∂Ω, Tϕ N ) × K0 → C k,α (Ω, Tu N ) is an immersion. (c). Dg(ϕ, 0) : C k,α (∂Ω, Tϕ N ) × K0 → K0 is a submersion. (d). For every ε > 0, there is a neighborhood W of u in C k,α (Ω, N ) such that each harmonic map v in W equals F (v|∂Ω, κ) for some κ ∈ K0 with kκkk,α < ε. (e). U and W can be chosen so that U = U ∩ g −1 (0) and W = W ∩ H are smooth submanifolds of C k,α (∂Ω, Tϕ N ) × K0 and C k,α (Ω, N ) respectively. F : U → W is an isomorphism and Π ◦ F (ψ, κ) = ψ. U has codimension dim K0 and its tangent space at (ϕ, 0) is T(ϕ,0) U = Ker D1 g(ϕ, 0) ⊕ K0 (f). For 0 < α0 < α, U and W can be replaced by U 0 and W 0 that are open, with respect to the corresponding 0 C k,α norms, in C k,α (∂Ω, N ) and C k,α (Ω, N ), respectively.
Remark. It is perhaps useful to check the analogue of this theorem in the well-known elementary case of geodesic arcs u : [0, 1] → N . Here the association of the initial position and initial velocity of a geodesic gives a global diffeomorphism between H and the tangent bundle of N . As above, each geodesic u ∈ H has a neighborhood diffeomorphic to a 2n-dimensional submanifold of a neighborhood of (0, u(0), u(1)) in K0 × N × N . Proof of Theorem 6.2. Proof of (a): Construction of F and g. Denote K0⊥ = [K0⊥ ] ∩ C k−2,α (Ω, Tu N ) and K00 = K0⊥ ∩ k,α C0 (Ω, Tu N ) and PK0⊥ : L2 (Ω, Tu N ) → K0⊥ the orthogonal projection. For ψ ∈ C k,α (∂Ω, N ), let ψ¯ be the unique function satisfying ∆ψ¯ = 0, on Ω, (6.2.1) ψ¯ = ψ − ϕ, on ∂Ω. ¯ the projection to Tu N . Clearly, Φ(ϕ) = 0, and Φ(ψ) ∈ C k,α (Ω, Rp ). By Schauder’s Let Φ(ϕ) = Pu ◦ ψ, estimates, if ψ is C k,α (∂Ω, N ) close to ϕ, then Φ(ψ) ∈ C k,α (Ω, Rp ) is close to 0. Define Θ : C k,α (∂Ω, N ) × K0 × K00 → K0⊥ , (6.2.2)
Θ(ψ, κ, η) =PK0⊥ ◦ Pu ◦ H[π(u + Φ(ψ) + κ + η)].
Then from Proposition 6.1, we have D3 Θ(ϕ, 0, 0) = PK0⊥ ◦ Pu ◦ DH|K00 = Ju |K00 , which is isomorphism. By the impicit function theorem, there are neighborhoods U = U1 × U2 of (ϕ, 0) in C k,α (∂Ω, N ) × K0 , V of 0 in K00 , and a smooth map Q : U ⊆ C k,α (∂Ω, N ) × K0 → K00 such that for (ψ, κ) ∈ U , η = Q(ψ, κ) is the unique solution in V of Θ(ψ, κ, η) = 0. Let (6.2.4)
F (ψ, κ) = π(u + Φ(ψ) + κ + Q(ψ, κ)), g(ψ, κ) = PK0 ◦ Pu ◦ H(F (ψ, κ)). 23
Then (6.2.5)
Θ(ψ, κ, Q(ψ, κ)) = PK0⊥ ◦ Pu ◦ H[F (ψ, κ))] = 0. g(ψ, κ) = Pu ◦ H(F (ψ, κ)),
.
Now that F and g satisfy (a) follows from their definitions and (4.7.3). Proof of (b). Let κ ∈ K0 . By definition of F , (6.2.6)
D2 F (ϕ, 0)κ = κ + D2 Q(ϕ, 0)κ.
Differentiating PK0⊥ ◦ Pu ◦ H[π(u + tκ + Q(ϕ, tκ))] = 0 for t at t = 0, we get 0 = hκ + D2 Q(ϕ, 0)κ, Ju i = hD2 Q(ϕ, 0)κ, Ju i. So D2 Q(ϕ, 0)κ ∈ K0 . On the other hand, the image of Q is in K0⊥ , therefore D2 Q(ϕ, 0)κ ∈ K0 ∩ K0⊥ = {0}. Thus D2 F (ϕ, 0)κ = κ. To show that DF (ϕ, 0) is an immersion, let (ξ, κ) ∈ C k,α (∂Ω, Tϕ N ) × K0 , then (6.2.7)
DF (ϕ, 0)(ξ, κ) = ξ∗ + κ,
where ξ∗ = D1 F (ϕ, 0)ξ.
Now look at the following sequence C k,α (∂Ω, Tϕ N ) × K0
DF (ϕ,0)
−→
C k,α (Ω, Tu N ) Pr
Λ
1 −→ C k,α (∂Ω, Tϕ N ) × C0k,α (Ω, Tu N ) −→ C k,α (∂Ω, Tϕ N ) × K0 ,
where Λ ξ = (ξ|∂Ω, ξ − ξ∗ ) is an isomorphism, as one can check easily. The composition P r1 ◦ Λ ◦ DF (ϕ, 0) = id, so DF (ϕ, 0) is an immersion [L, II,2]. Proof of (c). It suffices to show that Dg(ϕ, 0) : C k,α (∂Ω, Tϕ N ) × K0 → K0 is surjective. First note that for (ξ, κ) ∈ C k,α (∂Ω, Tϕ N ) × K0 , by definition of g and (6.2.7), (6.2.8)
Dg(ϕ, 0)(ξ, κ) = D1 g(ϕ, 0)ξ = Ju ξ∗ .
Secondly note that ∂κ : κ ∈ K0 } ∂n is a subspace of C k,α (∂Ω, Tϕ N ) of dimension dim(K0 ). This is because if 0 6= κ ∈ K0 , then k|∂Ω = 0 implies ∂κ ∂κ ∈ C k,α (∂Ω, Tϕ N ), and Calderon’s uniqueness theorem implies that 0 6= ([C1], [ML2, Prop.3.2].). ∂n ∂n Thus to show Dg(ϕ, 0) is surjective, it is enough to show that (6.2.9)
Vu|∂Ω = {
D1 g(ϕ, 0) : Vu|∂Ω → K0 is one-to-one. Indeed, by (6.2.8), D1 g(ϕ, 0)
∂κ = Ju η, ∂n
η=
∂κ ∂n
∗
.
∂κ ∂κ Thus, if D1 g(ϕ, 0) = 0, then η is a Jacobi field with η = on ∂Ω. By (1.6.1) and symmetry in η and ∂n ∂n κ, we will have Z Z Z Z ∂κ ∂κ ·η = | |2 , 0= −κ · Ju η = −Ju κ · η + ∂Ω ∂n Ω Ω ∂Ω ∂n ∂κ ≡ 0. So D1 g(ϕ, 0) is one-to-one. and so ∂n 24
Proof of (d). Suppose that v is a smooth harmonic map close to u. Let λ(x) ∈ Tu(x) N be the unique vector such that π(u(x) + λ(x)) = v(x). Then λ ∈ C k,α (Ω, Tu N ). Let h = Φ(ψ) − λ. Then (6.2.10)
v = π(u + Φ(ψ) + h)
Let κ = PK0 h, η = h − κ ∈ K0⊥ . Then (6.2.11)
v = π(u + Φ(ψ) + κ + η).
From the fact that v is harmonic, we have 0 = PK0⊥ ◦ Pu ◦ H[π(u + Φ(ψ) + κ + η)] = Θ(ψ, κ, η), which has a unique solution η = Q(ψ, κ). Thus by (6.2.11) and the definition of F , v = (F (ψ, κ)). Proof of (e). That Dg(ϕ, 0) is an immersion implies that g −1 (0) is a smooth submanifold near (ϕ, 0) of codimension dim(K0 ) [L, II,2]. By (b), (d) and again [L], we may choose U and W suitably small so that U = U ∩ g −1 (0), F (U) and W are all smooth manifolds, and F : U → W is an isomorphism. By (6.2.8), the tangent space to U at (ϕ, 0) is (6.2.12)
T(ϕ,0) U = Ker D1 g(ϕ, 0) ⊕ K0 . 0
Proof of (f ). Noting that u ∈ C k,α (Ω, N ) ⊆ C k,α (Ω, N ), we may apply (a) and (b) of the Theorem to u 0 with C k,α (Ω, N ) replaced by C k,α (Ω, N ). In this way, we get the corresponding neighborhoods U 00 of u|∂Ω 0 0 in C k,α (∂Ω, N ), W 00 of u in C k,α ;0 (Ω0 , N ) and a smooth map 0
0
F 0 : U 00 ⊆ C k,α (∂Ω, N ) → C k,α (Ω, N ). We first show F 0 : U 00 ∩ C k,α (∂Ω, N ) → C k,α (Ω, N ) is smooth and F 0 |U 00 ∩ U = F |U 00 ∩ U . 0 Indeed, if ψ ∈ U 00 ∩ C k,α (∂Ω, N ), then F 0 (ψ) ∈ C k,α (Ω, N ) is harmonic, that is, it satisfies
H(F 0 (ψ)) = ∆F 0 (ψ) − A(F 0 (ψ))(dF 0 (ψ), dF 0 (ψ)) = 0 on Ω, F 0 (ψ) = ψ on ∂Ω. 0
Since A(F 0 (ψ))(dF 0 (ψ), dF 0 (ψ)) ∈ C k−1,α (Ω, Rp ) ⊆ C k−2,α (Ω, Rp ) (see 1.2 (e)) and ψ ∈ C k,α (∂Ω, N ), it follows from Theorem 1.10 that F 0 (ψ) ∈ C k,α (Ω, N ). Furthermore, it follows from definitions that Q0 (ψ) = Q(ψ) on U 00 ∩ U , and then, F = F 0 on U 00 ∩ U . This also implies the smoothness of F 0 with respect to C k,α norms. Now we see that one may replace the F in (a) and (b) by F 0 , whose domain U 00 ∩ C k,α (∂Ω, N ) is open in C k,α (∂Ω, N ) with respect to norm k · kk,α0 . To end the proof of (c), set U 0 = U 00 ∩ C k,α (∂Ω, N ) and W 0 = W 00 ∩ C k,α (Ω, N ). Denote K = K(Ju ) = {κ ∈ C k,α (Ω, Tu N ) : Ju κ = 0}, and K0 = K0 (Ju ) = {κ ∈ C0k,α (Ω, Tu N ) : Ju κ = 0}. Again, a κ ∈ K is called a Jacobi field. We have 6.3. Theorem. ξ ∈ C k,α (Ω, Tu N ) is a Jacobi field to u ∈ H if and only if ξ is the initial velocity vector field of a one-parameter family of harmonic maps. 25
Proof : Suppose ξ ∈ K(Ju ). Let ξ∗ =: D1 F (u|∂Ω, 0)(ξ|∂Ω). Then by Theorem 6.2 (a), ξ∗ |∂Ω = ξ|∂Ω, and we now show ξ∗ ∈ K(Ju ). Indeed, by differentiating the identity 0 = PK0⊥ ◦ Pu ◦ H[F (π(u|∂Ω + tξ|∂Ω), 0))], we get 0 = PK0⊥ hξ∗ , Pu ◦ DH(u)i = PK0⊥ (Ju ξ∗ ). On the other hand Ju ξ∗ = Ju (ξ∗ − ξ) ∈ Im Ju = K0⊥ . So Ju ξ∗ = 0. Now let κ = ξ − ξ∗ , then κ ∈ K0 = K0 (Ju ). From (6.2.8), (6.2.12) we know that (ξ|∂Ω, κ) is in the tangent space to g −1 (0) at (u|∂Ω, 0). It follows from Theorem 6.2 (e) that there is a one-parameter family (ψt , κt ) in C k,α (∂Ω, N ) × K0 satisfying (ψ0 , κ0 ) = (u|∂Ω, 0), d (ψt , κt ) = (ξ|∂Ω, κ) dt t=0 such that g(ψt , κt ) = 0. Thus Theorem 6.2 (a) implies that F (ψt , κt ) is a one-parameter family of harmonic maps, which has initial velocity vector field: d F (ψt , κt ) = ξ∗ + κ = ξ. dt t=0 d d Conversely, suppose ξ = dt u with ut ∈ H and u0 = u. Then H(ut ) = 0 implies Ju ξ = dt H(ut ) = t=0 t t=0 0. That is, ξ ∈ K(Ju ). 6.4. Theorem (Global Structure). (a). H is a smooth Banach manifold modelled on C k,α (∂Ω, N ), with a countable cover of coordinate charts given in Theorem 6.2 (e) and (f ). ∂κ (b). The tangent space to H at u ∈ H is Tu H = K(Ju ). Vu|∂Ω = { : κ ∈ K0 (Ju )} is a subspace of ∂n k,α C (∂Ω, Tu|∂Ω N ) with dim (Vu|∂Ω ) = dim (K0 (Ju )) and perpendicular to DΠ(Tu H), where Π : H → C k,α (∂Ω, N ) in the projection. (c). The map Π : H → C k,α (∂Ω, N ) is Fredholm of index 0 with Ker DΠ(u) = K0 (Ju ) . So DΠ(u) is onto if and only if K0 (Ju ) = {0}. Proof : The set A of all such pairs (W, Γ) (Γ = F −1 ) in Theorem 6.2 (e) (f) form an atlas for H. To verify the smoothness of the transition maps, let (W1 , Γ1 ), (W2 , Γ2 ) be two charts corresponding to F1 and F2 , then −1 Γ1 ◦ Γ−1 |Γ2 (W1 ∩W2 ) , 2 |Γ2 (W1 ∩W2 ) = Γ1 ◦ F2 ◦ (Γ2 ◦ F2 ) is smooth. By the separability of H with respect to k · kk,α0 ,0 with 0 < α0 < α, A contains a countable subcover. This shows (a). The first part of (b) is just Theorem 6.3. To show the second part, let κ ∈ K0 (Ju ) and ξ ∈ Tu H. Then by symmetry of the second variational formula (1.6.1) for ξ, κ, and the facts Ju ξ = Ju κ = 0, we get Z Z ∂κ ∂ξ ·ξ = · κ = 0. ∂n ∂n ∂Ω ∂Ω That is, Vu|∂Ω ⊥ Tu H. As shown in (6.2.9), dim(Vu|∂Ω )= dim(K0 (Ju )). Now we show (c). Note that for κ ∈ Tu H, hκ, D(Π|H)(u)i = κ|∂Ω. Thus Ker D(Π|H)(u) = K0 (Ju ), and Im D(Π|H)(u) = K(Ju )/K0 (Ju ). By definition, Π|H is Fredholm of index 0. If D(Π|H)(u) is onto, then by (b), Vu|∂Ω = {0}, so K0 (Ju ) = {0}. The converse is obvious. 26
6.5. Definition. A boundary map ω ∈ C k,α (∂Ω, N ) is a regular value of Π if D(Π|H)(u) is surjective for all u ∈ H ∩ Π−1 {ω}. This means that each such u has no nonzero Jacobi fields that vanish on ∂Ω (Theorem 6.4 (c)). A nonregular value is also called singular. 6.6. Theorem. The set of singular values of Π is of first (Baire) category in C k,α (∂Ω, N ). Proof : For an open subset W ⊆ H, we define S(W), where S(W) = {ω ∈ C k,α (∂Ω, N ) : ω is a singular value of Π|W}. 0
Since H is separable with respect to C k,α norm (0 < α0 < α), to prove the theorem, it suffices to show 0 that, for each u ∈ H, there is a C k,α open neighborhood W of u in H, so that S(W) is closed and nowhere dense. For u ∈ H, let W be chosen as in Theorem 6.2 (e) such that W can be identified as a codimension dim(K0 ) submanifold of U × K0 . (K0 = K0 (Ju ).) Clearly S(W) is closed. Now we prove that it is also nowhere dense, that is, any ω ∈ S(W) can be approximated by an ϕ ∈ / S(W). Let V = W ∩ (expω V × K0 ), ∂κ : κ ∈ K0 (Ju )} and and expω is the exponential map (as in 1.2 (b)). Then V is ∂n a submanifold of expω V × K0 of dimension and codimension dim(K0 ). (cf. Theorem 6.4 (b).) Note that ϕ ∈ expω V is a regular value of Π|V if and only if it is a regular value of Π. Consider the Π : V ⊆ expω V × K0 → expω V . By Sard’s Theorem, there is a regular value z ∈ V of −1 expω Π that is arbitrarily close to 0. Thus ϕ = expω z is a regular value of Π|V, which can be arbitrarily close to ω. where V = Vu|∂Ω = {
As an application of Theorem 6.2 and Theorem 6.4, here we prove some finiteness and uniqueness results for harmonic maps analogous to [WB]. By Theorem 6.6, the following result can be considered as generic local finiteness. 6.7. Theorem. Suppose that ω is a regular value of Π. For any M > 0, there exists a C k,α neighborhood U of ω so that each ϕ ∈ U serves as boundary data for only finitely many harmonic maps u ∈ C k,α (Ω, N ) with kukk,α ≤ M . The number of these harmonic maps is bounded independent of ϕ ∈ U . Proof : Let α0 and X be in the proof of Corollary 6.7. By Theorem 6.1 (a) each harmonic map u ∈ C k,α (Ω, N ) has a C k,α neighborhood Wu0 so that Π maps the set of harmonic maps in Wu0 diffeomorphically 0 0 onto a C k,α neighborhood U 0 of ω in C k,α (∂Ω, N ). Since X is C k,α ;0 compact, Xω =: X ∩ Π−1 {ω} is finite, 0 V = ∩u∈Xω Uu0 is C k,α open, and δ = distC k,α0 Π−1 {ω}, (X \ Π−1 (V )) ∩ ∪u∈Xω Wu0 0
is positive. So we may choose a C k,α open neighborhood U about ω in V \ Π X ∩ {u : distC k,α0 (u, Π−1 {ω}) ≥ δ} . Let ϕ ∈ U . As in the proof of 6.7, each harmonic v ∈ C k,α (Ω, N ) with v|∂Ω = ϕ and kvkk,α ≤ M must belong to Xϕ = X ∩ Π−1 {ϕ}. Thus the number of such v’s is bounded by card(Xϕ ) ≤ card(Xω ). 27
As in [WB] we say that a subset A of a Banach manifold X is of codimension c if A ⊆ ∪∞ i=1 Πi (Mi ) ki where Mi is a codimension c + ki submanifold of X × R and Πi is the projection map of X × Rki onto X . On the space C k,α (∂Ω, N ), there are two other notions available: measure 0 (cf. [MF2][ML2]) and first category, both are weaker than codimension 1 ([WB, 1.7]). In [ML2], a uniqueness result for energy minimizers is proved in the sense of measure 0. 6.8. Theorem.. The set of all regular values of Π in C k,α (∂Ω, N ) that serves as boundary value of two or more harmonic maps in C k,α (Ω, N ) having the same energy is a codimension 1 subset of C k,α (∂Ω, N ). Proof : First note that the set of regular values of Π is an open and dense subset of C k,α (∂Ω, N ). Denote C = {(u1 ,u2 ) : u1 , u2 ∈ C k,α (Ω, N ) are harmonic maps with u1 |∂Ω = u2 |∂Ω being regular, u1 6= u2 and E(u1 ) = E(u2 )}. Let (u1 , u2 ) ∈ C with ui |∂Ω = ϕ. For i = 1, 2, let Fi be the maps in Theorem 6.1 corresponding to ui , 0 defined on neighborhood Ui of ϕ in C k,α (∂Ω, N ), which can be chosen to be open in the C k,α norm. Let U = U1 ∩ U2 and consider the function G : U ⊆ C k,α (∂Ω, N ) → R defined by
Z
Z |∇F1 (ψ)|2 −
G(ψ) = Ω
|∇F2 (ψ)|2 . Ω
For any ψ ∈ U , the map vi = Fi (ψ) is harmonic (i=1, 2), and for h ∈ Tu U = C k,α (∂Ω, Tϕ N ), by first variational formula 1.4.1, (6.7.1)
d hh, DG(ϕ)i = G(π(ψ + th)) = dt t=0
Z h· ∂Ω
∂u2 ∂u1 − . ∂n ∂n
Since u1 6= u2 , by a uniqueness continuation theorem in [ML2, Proposition 6.2], we have
∂u1 ∂u2 6= . This ∂n ∂n
implies that DG(ϕ) 6= 0. By implicit function theorem, G−1 (0) ∩ U is a submanifold of U of codimension 1. By Theorem 6.1 (a) there is a small δ(u1 , u2 ) > 0 such that if (v1 , v2 ) ∈ B(u1 , u2 ) =: {(v1 , v2 ) ∈ C : kvi − ui kk,α0 ;0 < δ(u1 , u2 ), i = 1, 2}, then vi = Fi (ψ) and ψ = v1 |∂Ω = v2 |∂Ω ∈ U . Thus by the definitions of G, ψ belongs to the family Σ(u1 , u2 ) = G−1 (0) ∩ U , which is of codimension 1 in C k,α (∂Ω, N ). Then the separability of C k,α (Ω, N ) with respect to norm k · kk,α0 ,0 implies that the open cover {B(u1 , u2 ) : (u1 , u2 ) ∈ C} of C has a countable subcover {B(ui1 , ui2 ) : (ui1 , ui2 ) ∈ C, i = 1, 2, 3, . . .}. Thus i i the relevant set of nonunique boundary data is contained in ∪∞ i=1 Σ(u1 , u2 ), which is of codimension 1. Because a subset of C k,α (∂Ω, N ) of codimension 1 is of first category, we may combine Theorems 6.6 and 6.8 to obtain the following: 6.9. Corollary. The set of all ϕ ∈ C k,α (∂Ω, N ) that serves as boundary value of two or more harmonic maps in C k,α (Ω, N ) having the same energy is of first category in C k,α (∂Ω, N ). REFERENCES 28
[ADN] S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solution of partial elliptic differential equations satisfying general boundary conditions II. Comm. Pure Appl. Math. 17 (1964), 35-92. [BA] A. Blades, Stability and uniqueness properties of the equator map from a ball into an ellipsoid, Math. Zeit. (1984) [BCL] H. Brezis and J.-M. Coron and E. H. Lieb, Harmonic maps with defects, Comm. Math. Phy., 107 (1986) 649-705. [C] A.P. Calder`on. Uniqueness in the Cauchy problems for partial differential equations. Amer. J. Math. 80 (1958), 16-35. [CHS] L. Caffarelli, R. Hardt and L. Simon, Minimal surfaces with isolated singularities. Manuscripta Math. 48(1984) 1-18. [DN] A. Douglis and L. Nirenberg, Interior estimates for elliptic systems of partial differential equations. Comm. Pure Appl. Math. 8.(1955), 503–538. [EL] J. Eells and L. Lemaire, Another report on harmonic maps, Bull. London Math. Soc. 20(1988), 385–524. [F] H. Federer, “Geometric Measure Theory”, Springer-Verlag, 1969. [GT] D. Gilbarg and N. Trudinger, “Elliptic Partial Differential Equations of Second Order”, Springer–Verlag, New York, Berlin, Heildelberg, 1983. [HKL] R. Hardt, D. Kinderlehrer and F. H. Lin,Existence and partial regularity of static liquid crystal configurations. Comm. Math. Physics, 105(1986), 547–570. [HL1] R. Hardt and F.H. Lin, Mappings minimizing the Lp norm of the gradient. Comm. Pure Appl. Math. 40(1987), 555–588. [HL2] R. Hardt and F.H. Lin, Stability of singularities of minimizing harmonic maps. J. Diff. Geom. 29(1987), 113–123. [HS] R. Hardt and L. Simon, Area minimizing hypersurfaces with isolated singularities, J. Reine. Angew. Math. 362 (1985), 102-129. [H] S. Hildebrandt, Harmonic mappings of Riemaniann manifolds, in Lecture Notes in Mathematics 1161, Springer-Verlag, 1985, pp 1-117. [JK] W. J¨ager and H. Kaul, Rotationally symmetric harmonic maps from a ball into a sphere and the regularity problem for weak solutions of elliptic systems. J. Reine Angew. Math. 343 (1983) 146-161. [J] J. Jost, “Harmonic mappings between Riemannian manifolds”, Proc. Centre Math. Analysis, 1983. [L] S. Lang, “Differential Manifolds”, Adison-Wesley, Reading, Massachusetts. x [LF] F.H. Lin, A remark on the map |x| , Acad. Sci. Paris 305, I (1987), 529-531. [MS] R. Macintosh and L. Simon, Perturbing away singularities of harmonic maps, Manuscripta Mathematica, 67 (1990) 113-124. [MF1] F. Morgan, Almost every curve in R3 bounds a unique area minimizing surface. Invent. Math. 45(1978), 253–297. [MF2] F. Morgan, Measure on space of surfaces. Arch. Rational Mech. Anal. 78(1982), 335–359. [MF3] F. Morgan, Generic uniqueness for hypersurfaces minimizing the integrand with constant coefficients. Indiana U. Math. J. 30(1981), 29–45. [MC] C. B. Morrey, Jr., “Multiple Integrals in the Calculus of Variations”, Springer-Verlag, 1966. [ML1] L. Mou, Harmonic maps with prescribed finite singularities. Comm. in P. D. E. Vol. 14, 11(1989), 1509–1540. [ML2] L. Mou, Uniqueness of energy minimizing maps for almost all smooth boundary data. Indiana Univ. Math. Jour. Vol. 40, No 1 (1991) 363-392. [SU1] R. Schoen and K. Uhlenbeck, A regularity theory for harmonic maps. J. Diff. Geom. 17(1982), 307–335. 29
[SL] L. Simon, Isolated singularities for extrema of general variational problems. Lecture Notes in Mathematics 1161, Springer-Verlag, 1985. [SN1] N. Smale, Minimal surfaces with many isolated singularities Ann. of Math. 130(1989), 603–642. [SN2] N. Smale, An equivariant construction of minimal surfaces with nontrivial singular sets, Indiana Univ. Math. Jour. 40, No.2 (1991)595-616. [SN3] N. Smale, Geometric P.D.E.’s with isolated singularities. Preprint. [SRT] R. T. Smith, The second variation for harmonic mappings, Proc. Amer. Math. Soc. 47 (1975) 229-236. [T] M. Taylor, “Pseudodifferential Operators”, Princeton University Press, 1981. [WB] B. White, The space of m-dimensional surfaces that are stationary for a parametric elliptic functional, Indiana Univ. Math. Jour. Vol. 36, 3(1987), 567–602.
*Research partially supported by the National Science Foundation Mathematics Department, Rice University, Houston, TX 77251
Department of Mathematics, University of Southern California, Los Angeles, CA 90089
30
